The sparsity of random graphs as well as randomly perturbed trees was used in showing that these families have w.h.p. bounded expansion.Bounded expansion should not be confused with the edge expansion of a graph. For a precise definition please see [1]}, [2]}. [1]}, [4]} These results are incomparable with our treewidth results: it is known that graphs with bounded maximum degree have bounded expansion and that \(G(n,d/n)\)  has bounded expansion w.h.p. [1]}, [2]} In contrast, there exist 3-regular graphs with linear treewidth and as previously mentioned the treewidth of \(G(n,d/n)\)  is \(\Omega (n)\) .
